Question: Simplify; express your answer in exponential form. Assume $a\neq 0, t\neq 0$. $\dfrac{{(a^{-4})^{-3}}}{{(a^{-3}t^{5})^{-5}}}$
Answer: To start, try working on the numerator and the denominator independently. In the numerator, we have ${a^{-4}}$ to the exponent ${-3}$ . Now ${-4 \times -3 = 12}$ , so ${(a^{-4})^{-3} = a^{12}}$ In the denominator, we can use the distributive property of exponents. ${(a^{-3}t^{5})^{-5} = (a^{-3})^{-5}(t^{5})^{-5}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(a^{-4})^{-3}}}{{(a^{-3}t^{5})^{-5}}} = \dfrac{{a^{12}}}{{a^{15}t^{-25}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{12}}}{{a^{15}t^{-25}}} = \dfrac{{a^{12}}}{{a^{15}}} \cdot \dfrac{{1}}{{t^{-25}}} = a^{{12} - {15}} \cdot t^{- {(-25)}} = a^{-3}t^{25}$.